The existence of positive solutions for a critically coupled Schrödinger system in a ball of [formula omitted]

Abstract

In this paper, we consider the following coupled Schrödinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem{−Δu+λ1u=μ1u5+βu2v3,x∈Ω,−Δv+λ2v=μ2v5+βv2u3,x∈Ω,u=v=0,x∈∂Ω, where Ω is a ball in R3, −λ1(Ω)<λ1,λ2<−14λ1(Ω), μ1,μ2>0 and β>0. Here λ1(Ω) is the first eigenvalue of −Δ with Dirichlet boundary condition in Ω. We show that the problem has at least one positive solution for all 0<β<2μ1μ2. In particular, when λ1=λ2, we prove the existence of positive synchronized solutions for all β>0.